One has analogous definitions for analytic invariants associated to the \(\Sigma _0\) -imprimitive \(p\) -adic L-function \(\mathcal {L}^{\Sigma _0}(E,T)\) . For \(l\in \Sigma _0\) , it is the case that \(\mathcal {P}_l\)  generates the characteristic ideal of the Pontryajin dual of \(\mathcal {H}_l(\mathbb {Q}^{cyc})\)  [1]}. It follows from the relation \(\ref {equationscriptL}\)  that
\(\lambda _{E,\Sigma _0}^{anal}=\lambda _{E}^{anal}+\sum _{l\in \Sigma _0} \sigma _E^{(l)}.\) 
